Question

In problem $\mathbf{1}\mathbf{-}\mathbf{6}$ , determine whether the given differential equation is separable ${\mathit{s}}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{d}\mathbf{s}}{\mathbf{d}\mathbf{t}}\mathbf{=}\frac{\mathbf{s}\mathbf{+}\mathbf{1}}{\mathbf{s}\mathbf{t}}$.

Short answer

The given differential equation is not separable.

Step-by-step solution

Concept of Separable Differential Equation

A first-order ordinary differential equation $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{f}\left(x,y\right)$is referred to as separable if the function in the right-hand side of the equation is expressed as a product of two functions $\mathit{g}\left(x\right)$that is a function of $\mathit{x}$alone and $\mathit{h}\left(y\right)$that is a function of $\mathit{y}$alone.

Mathematically, the equation $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{f}\left(x,y\right)$is separable when role="math" localid="1654781548639" $\mathit{f}\left(x,y\right)\mathbf{=}\mathit{g}\left(x\right)\mathbf{+}\mathit{h}\left(y\right)$.

Subtracting  s2 from both side in the given equation.

Given Equation$s^2+\frac{d\mathrm{s}}{dt}=\frac{s+1}{st}..........\left(1\right)$

Subtracting from both side in the equation

$\frac{ds}{dt}=\frac{s+1}{st}-s^2$

Simplifying the above equation

$$\begin{gathered} \frac{ds}{dt}=\frac{s+1-s^2\left(st\right)}{st} \\ \frac{ds}{dt}=\frac{s+1-s^{3}t}{st} \end{gathered}$$

We cannot separate the above expression about terms.

Therefore, the given differential equation is not separable.